Proposition 1.1 (Characterization of phase-locked states, [1]}) 
A solution \(\lbrace X_i\rbrace _{i=1}^N\)  to (REF ) is a phase-locked state if and only if \(X_i(t)=X_i^\infty \exp (\Lambda t)\) , where \(\exp \)  is the Lie group exponential and \(X_i^\infty \in G\) , \(i=1,\cdots ,N\) , and \(\Lambda \in \mathfrak {g}\)  satisfy
\(\operatorname{Ad}_{X_i^\infty }\Lambda = H_i+\frac{\kappa }{N}\sum _{j=1}^N\phi (X_j^\infty (X_i^\infty )^{-1}),\quad i=1,\cdots ,N,\) 
